Functional Differential Equations: Advances and Applications

By Constantin Corduneanu, Yizeng Li and Mehran Mahdavi

Features new results and up-to-date advances in modeling and solving differential equations

Introducing the various classes of functional differential equations, Functional Differential Equations: Advances and Applications presents the needed tools and topics to study the various classes of functional differential equations and is primarily concerned with the existence, uniqueness, and estimates of solutions to specific problems. The book focuses on the general theory of functional differential equations, provides the requisite mathematical background, and details the qualitative behavior of solutions to functional differential equations.

The book addresses problems of stability, particularly for ordinary differential equations in which the theory can provide models for other classes of functional differential equations, and the stability of solutions is useful for the application of results within various fields of science, engineering, and economics. Functional Differential Equations: Advances and Applications also features:

• Discussions on the classes of equations that cannot be solved to the highest order derivative, and in turn, addresses existence results and behavior types

• Oscillatory motion and solutions that occur in many real-world phenomena as well as in man-made machines

• Numerous examples and applications with a specific focus on ordinary differential equations and functional differential equations with finite delay

• An appendix that introduces generalized Fourier series and Fourier analysis after periodicity and almost periodicity

• An extensive Bibliography with over 550 references that connects the presented concepts to further topical exploration

Functional Differential Equations: Advances and Applications is an ideal reference for academics and practitioners in applied mathematics, engineering, economics, and physics. The book is also an appropriate textbook for graduate- and PhD-level courses in applied mathematics, differential and difference equations, differential analysis, and dynamics processes.

CONSTANTIN CORDUNEANU, PhD, is Emeritus Professor in the Department of Mathematics at The University of Texas at Arlington, USA. The author of six books and over 200 journal articles, he is currently Associate Editor for seven journals; a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Romanian Academy; and past president of the American Romanian Academy of Arts and Sciences.

YIZENG LI, PhD, is Professor in the Department of Mathematics at Tarrant County College, USA. He is a member of the Society for Industrial and Applied Mathematics.

MEHRAN MAHDAVI, PhD, is Professor in the Department of Mathematics at Bowie State University, USA. The author of numerous journal articles, he is a member of the American Mathematical Society, Society for Industrial and Applied Mathematics, and the Mathematical Association of America.

• Language: English
• Publisher: Wiley
• Release date: Mar 30, 2016
• ISBN: 9781119189497

Book preview

1

INTRODUCTION, CLASSIFICATION, SHORT HISTORY, AUXILIARY RESULTS, AND METHODS

Generally speaking, a functional equation is a relationship containing an unknown element, usually a function, which has to be determined, or at least partially identifiable by some of its properties. Solving a functional equation (FE) means finding a solution, that is, the unknown element in the relationship. Sometimes one finds several solutions (solutions set), while in other cases the equation may be deprived of a solution, particularly when one provides the class/space to which it should belong.

Since a relationship could mean the equality, or an inequality, or even the familiar belongs to, designated by , , or , the description given earlier could also include the functional inequalities or the functional inclusions, rather often encountered in the literature. Actually, in many cases, their theory is based on the theory of corresponding equations with which they interact. For instance, the selection of a single solution from a solution set, especially in case of inclusions.

In this book we are mainly interested in FEs, in the proper/usual sense. We send the readers to adequate sources for cases of related categories, like inequalities or inclusions.Functional Differential Equations: Advances and Applications, First Edition.Constantin Corduneanu, Yizeng Li and Mehran Mahdavi© 2016 John Wiley & Sons, Inc. Published 2016 by John Wiley & Sons, Inc.

1.1 CLASSICAL AND NEW TYPES OF FEs

The classical types of FEs include the ordinary differential equations (ODEs), the integral equations (IEs) of Volterra or Fredholm and the integro-differential equations (IDEs). These types, which have been thoroughly investigated since Newton’s time, constitute the classical part of the vast field of FEs, or functional differential equations (FDEs).

The names Bernoulli, Newton, Riccati, Euler, Lagrange, Cauchy (analytic solutions), Dini, and Poincaré as well as many more well-known mathematicians, are usually related to the classical theory of ODE. This theory leads to a large number of applications in the fields of science, engineering, economics, in cases of the modeling of specific problems leading to ODE.

A large number of books/monographs are available in the classical field of ODE: our list of references containing at least those authored by Halanay [237], Hale [240], Hartman [248], Lefschetz [323], Petrovskii [449], Sansone and Conti [489], Rouche and Mawhin [475], Nemytskii and Stepanov [416], and Coddington and Levinson [106].

Another classical type of FEs, closely related to the ODEs, is the class of IEs, whose birth is related to Abel in the early nineteenth century. They reached an independent status by the end of nineteenth century and the early twentieth century, with Volterra and Fredholm. Hilbert is constituting his theory of linear IEs of Fredholm’s type, with symmetric kernel, providing a successful start to the spectral theory of completely continuous operators and orthogonal function series.

Classical sources in regard to the basic theory of integral equations include books/monographs by Volterra [528], Lalesco [319], Hilbert [261], Lovitt [340], Tricomi [520], Vath [527]. More recent sources are Corduneanu [135], Gripenberg et al. [228], Burton [80, 84], and O’Regan and Precup [430].

A third category of FEs, somewhat encompassing the differential and the IEs, is the class of IDEs, for which Volterra [528] appears to be the originator. It is also true that E. Picard used the integral equivalent of the ODE , under initial condition , Cauchy’s problem, namely

obtaining classical existence and uniqueness results by the method of successive approximations.

A recent reference, mostly based on classical analysis and theories of DEs and IEs, is Lakshmikantham and R. M. Rao [316], representing a rather comprehensive picture of this field, including some significant applications and indicating further sources.

The extended class of FDEs contains all preceding classes, as well equations involving operators instead of functions (usually from R into R). The classical categories are related to the use of the so-called Niemytskii operator, defined by the formula , with or in an interval of R, while in the case of FDE, the right-hand side of the equation

implies a more general type of operator F. For instance, using Hale’s notation, one can take , where , represents a restriction of the function x(t), to the interval . This is the finite delay case. Another choice is

where V represents an abstract Volterra operator (see definition in Chapter 2), also known as causal operator.

Many other choices are possible for the operator F, leading to various classes of FDE. Bibliography is very rich in this case, and exact references will be given in the forthcoming chapters, where we investigate various properties of equations with operators.

The first book entirely dedicated to FDE, in the category of delay type (finite or infinite) is the book by A. Myshkis [411], based on his thesis at Moscow State University (under I. G. Petrovskii). This book was preceded by a survey article in the Uspekhi Mat. Nauk, and one could also mention the joint paper by Myshkis and Eĺsgoĺtz [412], reviewing the progress achieved in this field, due to both authors and their followers. The book Myshkis [411] is the first dedicated entirely to the DEs with delay, marking the beginnings of the literature dealing with non-traditional FEs.

The next important step in this direction has been made by N. N. Krasovskii [299], English translation of 1959 Russian edition. In his doctoral thesis (under N. G. Chetayev), Krasovskii introduced the method of Liapunov functionals (not just functions!), which permitted a true advancement in the theory of FDEs, especially in the nonlinear case and stability problems. The research school in Ekaterinburg has substantially contributed to the progress of the theory of FDEs (including Control Theory), and names like Malkin, Barbashin, and Krasovskii are closely related to this progress.

The third remarkable step in the development of the theory of FDE has been made by Jack Hale, whose contribution should be emphasized, in respect to the constant use of the arsenal of Functional Analysis, both linear and nonlinear. A first contribution was published in 1963 (see Hale [239]), utilizing the theory of semigroups of linear operators on a Banach function space. This approach allowed Hale to develop a theory of linear systems with finite delay, in the time-invariant framework, dealing with adequate concepts that naturally generalize those of ODE with constant coefficients (e.g., characteristic values of the system/equation). Furthermore, many problems of the theory of nonlinear ODE have been formulated and investigated for FDE (stability, bifurcation, and others (a.o.)). The classical book of Hale [240] appears to be the first in this field, with strong support of basic results, some of them of recent date, from functional analysis.

In the field of applications of FDE, the book by Kolmanovskii and Myshkis [292] illustrates a great number of applications to science (including biology), engineering, business/economics, environmental sciences, and medicine, including the stochastic factors. Also, the book displays a list of references with over 500 entries.

In concluding this introductory section, we shall mention the fact that the study of FDE, having in mind the nontraditional types, is the focus for a large number of researchers around the world: Japan, China, India, Russia, Ukraine, Finland, Poland, Romania, Greece, Bulgaria, Hungary, Austria, Germany, Great Britain, Italy, France, Morocco, Algeria, Israel, Australia and the Americas, and elsewhere.

The Journal of Functional Differential Equations is published at the College of Judea and Samaria, but its origin was at Perm Technical University (Russia), where N. V. Azbelev created a school in the field of FDE, whose former members are currently active in Russia, Ukraine, Israel, Norway, and Mozambique.

Many other journals are dedicated to the papers on FDE and their applications. We can enumerate titles like Nonlinear Analysis (Theory, Methods & Applications), published by Elsevier; Journal of Differential Equations; Journal of Mathematical Analysis and Applications, published by Academic Press; Differentsialuye Uravnenja (Russian: English translation available); and Funkcialaj Ekvacioj (Japan). Also, there are some electronic journals publishing papers on FDE: Electronic Journal of Qualitative Theory of Differential Equations, published by Szeged University; EJQTDE, published by Texas State University, San Marcos.

1.2 MAIN DIRECTIONS IN THE STUDY OF FDE

This section is dedicated to the description of various types of problems arising in the investigation of FDE, at the mathematical side of the problem as well as the application of FDE in various fields, particularly in science and engineering.

A first problem occurring in relationship with an FDE is the existence or absence of a solution. The solution is usually sought in a certain class of functions (scalar, vector, or even Banach space valued) and a priori limitations/restrictions may be imposed on it.

In most cases, besides the pure existence, we need estimates for the solutions. Also, it may be necessary to use the numerical approach, usually approximating the real values of the solution. Such approximations may have a local character (i.e., valid in a neighborhood of the initial/starting value of the solution, assumed also unique), or they may be of global type, keeping their validity on the whole domain of definition of the solution.

Let us examine an example of a linear FDE, of the form

(1.1)

with

a linear, casual continuous map, while . As shown in Corduneanu [149; p. 85], the unique solution of equation (1.1), such that , is representable by the formula

(1.2)

In (1.2), the Cauchy matrix is given, on , by the formula

(1.3)

where stands for the conjugate kernel associated to the kernel k(t, s), the latter being determined by the relationship

(1.4)

For details, see the reference indicated earlier in the text.

Formula (1.2) is helpful in finding various estimates for the solution x(t) of the initial value problem considered previously.

Assume, for instance, that the Cauchy matrix X(t, s) is bounded on by M, that is, ; hence , then (1.2) yields the following estimate for the solution x(t):

(1.5)

with and f continuous on [0, T]. We derive from (1.5) the estimate

(1.6)

which means an upper bound of the norm of the solutions, in terms of data.

We shall also notice that (1.6) keeps its validity in case , that is, we consider the problem on the semiaxis . This example shows how, assuming also , all solutions of (1.1) remain bounded on the positive semiaxis.

Boundedness of all solutions of (1.1), on the positive semiaxis, is also assured by the conditions , , and

The readers are invited to check the validity of the following estimate:

(1.7)

Estimates like (1.6) or (1.7), related to the concept of boundedness of solutions, are often encountered in the literature. Their significance stems from the fact that the motion/evolution of a man-made system takes place in a bounded region of the space. Without having estimates for the solutions of FDE, it is practically impossible to establish properties of these solutions.

One of the best examples in this regard is constituted by the property of stability of an equilibrium state of a system, described by the FDE under investigation. At least, theoretically, the problem of stability of a given motion of a system can be reduced to that of an equilibrium state. Historically, Lagrange has stated a result of stability for the equilibrium for a mechanical system, in terms of a variational property of its energy. This idea has been developed by A. M. Liapunov [332] (1857–1918), who introduced the method of an auxiliary function, later called Liapunov function method. Liapunov’s approach to stability theory is known as one of the most spectacular developments in the theory of DE and then for larger classes of FDE, starting with N. N. Krasovskii [299].

The comparison method, on which we shall rely (in Chapter 3), has brought new impetus to the investigation of stability problems. The schools created by V. V. Rumiantsev in Moscow (including L. Hatvani and V. I. Vorotnikov), V. M. Matrosov in Kazan, then moved to Siberia and finally to Moscow, have developed a great deal of this method, concentrating mainly on the ODE case. Also, V. Lakshmikantham and S. Leela have included many contributions in their treaty [309]. They had many followers in the United States and India, publishing a conspicuous number of results and developments of this method. One of the last contributions to this topic [311], authored by Lakshmikantham, Leela, Drici, and McRae, contains the general theory of equations with causal operators, including stability problems.

The comparison method consists of the simultaneous use of Liapunov functions (functionals), and differential inequalities. Started in its general setting by R. Conti [110], it has been used to prove global existence criteria for ODE. In short time, the use has extended to deal with uniqueness problems for ODE by F. Brauer [75] and Corduneanu [114, 115] for stability problems. The method is still present in the literature, with contributions continuing those already included in classical references due to Sansone and Conti [489], Hahn [235], Rouche and Mawhin [475], Matrosov [376–378], Matrosov and Voronov [387], Lakshimikantham and Leela [309], and Vorotnikov [531].

A historical account on the development of the stability concept has been accurately given by Leine [325], covering the period from Lagrange to Liapunov. The mechanical/physical aspects are emphasized, showing the significance of the stability concept in modern science. The original work of Liapunov [332] marks a crossroad in the development of this concept, with so many connections in the theory of evolutionary systems occurring in the mathematical description in contemporary science.

In Chapter 3, we shall present stability theory for ODE and FDE, particularly for the equations with finite delay. The existing literature contains results related to the infinite delay equations, a theory that has been originated by Hale and Kato [241]. An account on the status of the theory, including stability, is to be found in Corduneanu and Lakshmikantham [167]. We notice the fact that a theory of stability, for general classes of FDE, has not yet been elaborated. As far as special classes of FDE are concerned, the book [84] by T. Burton presents the method of Liapunov functionals for integral equations, by using modern functional analytic methods. The book [43] by Barbashin, one of the first in this field, contains several examples of constructing Liapunov functions/functionals.

The converse theorems in stability theory, in the case of ODE, have been obtained, in a rather general framework, by Massera [373], Kurzweil [303], and Vrkoč [532]. Early contributions to stability theory of ODE were brought by followers of Liapunov, (see Chetayev [103] and Malkin [356]). In Chapter 3, the readers will find, besides some basic results on stability, more bibliographical indications pertaining to this rich category of problems.

As an example, often encountered in some books containing stability theory, we shall mention here the classical result (Poincaré and Liapunov) concerning the differential system , , , and a continuous map. If we admit the commutativity condition

(1.8)

then the solution, under initial condition , can be represented by

(1.9)

From this representation formula one derives, without difficulty, the following results:

Stability of the solution the zero vector in Rn is equivalent to boundedness, on , of the matrix function .

Asymptotic stability of the solution is equivalent to the condition

(1.10)

Both statements are elementary consequences of formula (1.8). The definitions of various types of stability will be done in Chapter 3. We notice here that the already used terms, stability and asymptotic stability, suggest that the first stands for the property of the motion to remain in the neighborhood of the equilibrium point when small perturbations of the initial data are occurring, while the second term tells us that besides the property of stability (as intuitively described earlier), the motion is actually tending or approaching indefinitely the equilibrium state, when .

Remark 1.1

The aforementioned considerations help us derive the celebrated stability result, known as Poincaré–Liapunov stability theorem for linear differential systems with constant coefficients.

Indeed, if constant is an matrix, with real or complex coefficients, with characteristic equation , the unit matrix of type , then we denote by λ1, λ2, …, λk its distinct roots ( ). From the elementary theory of DEs with constant coefficients, we know that the entries of the matrix eA t are quasi-polynomials of the form

(1.11)

with pj(t), , some algebraic polynomials.

Since the commutativity condition (1.8) is valid when constant, there results that (1.10) can hold if and only if the condition

(1.12)

is satisfied. Condition (1.12) is frequently used in stability theory, particularly in the case of linear systems encountered in applications, but also in the case of nonlinear systems of the form

(1.13)

when f—using an established odd term—is of higher order with respect to x (say, for instance, ).

We will conclude this section with the discussion of another important property of motion, encountered in nature and man-made systems. This property is known as oscillation or oscillatory motion. Historically, the periodic oscillations (of a pendulum, for instance) have been investigated by mathematicians and physicists.

Gradually, more complicated oscillatory motions have been observed, leading to the apparition of almost periodic oscillations/vibrations. In the third decade of the twentieth century, Harald Bohr (1887–1951), from Copenhagen, constructed a wider class than the periodic one, called almost periodic.

In the last decade of the twentieth century, motivated by the needs of researchers in applied fields, even more complex oscillatory motions have emerged. In the books by Osipov [432] and Zhang [553, 554], new spaces of oscillatory functions/motions have been constructed and their applications illustrated.

In case of the Bohr–Fresnel almost periodic functions, a new space has been constructed, its functions being representable by generalized Fourier series of the form

(1.14)

with and , .

In the construction of Zhang, the attached generalized Fourier series has the form

(1.15)

with , , and , Q(R) denoting the algebra of polynomial functions of the form

(1.16)

and for ; , while denote arbitrary reals.

The functions (on R) obtained by uniform approximation with generalized trigonometric polynomials of the form

(1.17)

are called strong limit power functions and their space is denoted by .

A discussion of these generalizations of the classical trigonometric series and attached sum are presented in Appendix. The research work is getting more and more adepts, contributing to the development of this third stage in the history of oscillatory motions/functions.

In order to illustrate, including some applications to FDEs, the role of almost periodic oscillations/motions, we have chosen to present in Chapter 4 only the case of APr-almost periodic functions, , constituting a relatively new class of almost periodic functions, related to the theory of oscillatory motions. Their construction is given, in detail, in Chapter 4, as well as several examples from the theory of FDEs.

Concerning the first two stages in the development of the theory of oscillatory functions, the existing literature includes the treatises of Bary [47] and Zygmund [562]. These present the main achievements of the first stage of development (from Euler and Fourier, to contemporary researchers). With regard to the second stage in the theory of almost periodic motions/functions, there are many books/monographs dedicated to the development, following the fundamental contributions brought by Harald Bohr. We shall mention here the first books presenting the basic facts, Bohr [72] and Besicovitch [61], Favard [208], Fink [213], Corduneanu [129, 156], Amerio and Prouse [21], and Levitan [326], Levitan and Zhikov [327]. These references contain many more indications to the work of authors dealing with the theory of almost periodic motions/functions. They will be mentioned in Section 4.9.

As an example of an almost periodic function, likely the first in the literature but without naming it by its name, seems to be due to Poincaré [454], who dealt with the representations of the form

(1.18)

Supposing that the series converges uniformly to f on R (which situation can occur, for instance, when ), Poincaré found the formula for the coefficients ak, introducing simultaneously the concept of mean value of a function on R:

(1.19)

This concept was used 30 years later by H. Bohr, to build up the theory of almost periodic functions (complex-valued). The coefficients were given by the formula

(1.20)

1.3 METRIC SPACES AND RELATED CONCEPTS

One of the most frequent tools encountered in modern mathematical analysis is a metric space, introduced at the beginning of the twentieth century by Maurice Fréchet (in his Ph.D. thesis at Sorbonne). This concept came into being after G. Cantor laid the bases of the set theory, opening a new era in mathematics. The simple idea, exploited by Fréchet, was to consider a distance between the elements of an abstract set.

Definition 1.1

A set S, associated with a map , is called a metric space, if the following axioms are adopted:

, with only when ;

, ;

, .

Several consequences can be drawn from Definition 1.1. Perhaps, the most important is contained in the following definition:

Definition 1.2

Consider a sequence of elements/points . If

(1.21)

then one says that the sequence converges to x in S.

Then x is called the limit of the sequence.

It is common knowledge that the limit of a convergent sequence in S is unique.

Since the concept of a metric space has gained wide acceptance in Mathematics, Science and Engineering, we will send the readers to the book of Friedman [214] for further elementary properties of metric spaces and the concept of convergence.

It is important to mention the fact that the concept of convergence/limit helps to define other concepts, such as compactness of a subset . Particularly, the concept of a complete metric space plays a significant role.

Definition 1.3

The metric space (S, d) is called complete, if any sequence satisfying the Cauchy condition, "for each , there exists an integer , such that for , is convergent in (S, d)."

Definition 1.4

The metric space (S, d) is called compact, according to Fréchet, iff any sequence contains a convergent subsequence , that is, such that , for some .

Definition 1.4 leads easily to other properties of a compact metric space. For instance, the diameter of a compact metric space S is finite: . Also, every compact metric space is complete.

We rely on other properties of the metric spaces, sending the readers to the aforementioned book of Friedman [214], which contains, in a concise form, many useful results we shall use in subsequent sections of this book. Other references are available in the literature: see, for instance, Corduneanu [135], Zeidler [551], Kolmogorov and Fomin [295], Lusternik and Sobolev [343], and Deimling [190].

Almost all books mentioned already contain applications to the theory of FEs, particularly to differential equations and to integral equations. Other sources can be found in the titles referenced earlier in the text.

The metric spaces are a particular case of topological spaces. The latter represent a category of mathematical objects, allowing the use of the concept of limit, as well as many other concepts derived from that of limit (of a sequence of a function, limit point of a set, closure of a set, closed set, open set, a.o.)

If we take the definition of a topological space by means of the axioms for the family of open sets, then in case of metric spaces the open sets are those subsets A of the space S, defined by the property that any point x of A belongs to A, together with the ball of arbitrary small radius r, .

It is easy to check that the family of all open sets, of a metric space S, verifies the following axioms (for a topological space):

The union of a family of open sets is also an open set.

The intersection of a finite family of open sets is also an open set.

The space S and the empty set belong to the family of open sets.

Such a family, satisfying axioms 1, 2, and 3, induces a topology τ on S. Returning to the class of metric spaces, we shall notice that the couple (S, d) is inducing a topology on S and, therefore, any property of topological nature of this space is the product of the metric structure (S, d). The converse problem, to find conditions on a topological space to be the product of a metric structure, known as metrizability, has kept the attention of mathematicians for several decades of the past century, being finally solved. The result is known as the theorem of Nagata–Smirnov.

Substantial progress has been made, with regard to the enrichment of a metric structure, when Banach [39] introduced the new concept of linear metric space, known currently as Banach space.

Besides the metric structure/space (S, d), one assumes that S is a linear space (algebraically) over the field of reals R, or the field of complex numbers C. Moreover, there must be some compatibility between the metric structure and the algebraic one. Accordingly, the following system of axioms is defining a Banach space, denoted , with a map from S into , , called a norm.

S is a linear space over R, in additive notation.

S is a normed space, that is, there is a map, from S into , , satisfying the following conditions:

for , iff ;

, for , ;

for .

It is obvious that , , is a distance/metric on S.

(S, d), with d defined earlier, is a complete metric space.

Also, traditional notations for a Banach space, frequently encountered in literature, are or , in the latter case, the generic element of X being denoted by x.

The most commonly encountered Banach space is the vector space Rn (or Cn), the norm being usually defined by

and called the Euclidean norm. Another norm is defined by

.

Both norms mentioned previously lead to the same kind of convergence in Rn, because , . This is the usual convergence on coordinates, that is,

.

A special type of Banach space is the Hilbert space. The prototype has been constructed by Hilbert, and it is known as ℓ²(R), or ℓ²( ), space. This fact occurred long before Banach introduced his concept of space in the 1920s. The ℓ²-space appeared in connection with the theory of orthogonal function series, generated by the Fredholm–Hilbert theory of integral equations with symmetric kernel (in the complex case, the condition is . It is also worth mentioning that the first book on Hilbert spaces, authored by M. Stone [508], shortly preceded the first book on Banach space theory, Banach [39], 1932. The Banach spaces reduce to Hilbert spaces, in the real case, if and only if the rule of the parallelogram is valid:

(1.22)

Of course, the parallelogram involved is the one constructed on the vectors x and y as sides.

What is really specific for Hilbert spaces is the fact that the concept of inner product is defined for , as follows: it is a map from into R (or ), such that

;

, the value 0 leading to ;

, with , .

In the complex case, one should change to , remains the same, and must be changed accordingly.

If one starts with a Banach space satisfying condition (1.22), then the inner product is given by

(1.23)

Conditions , , and , stated in the text, can be easily verified by the product given by formula (1.23).

A condition verified by the inner product is known as Cauchy inequality, and it looks

(1.24)

It is easily obtained starting from the obvious inequality , which is equivalent to

, which, regarded as a quadratic polynomial in a, must take only nonnegative values. This would be possible only in case the discriminant is nonpositive, that is, , which implies (1.24). Using (1.24), prove that , for any .

In concluding this section, we will define another special case of linear metric spaces, whose metric is invariant to translations. These spaces are known as linear Fréchet spaces. We will use them in Chapter 2.

If instead of a norm, satisfying conditions II and III in the definition of a Banach space, we shall limit the imposed properties to , accepting the possibility that there may be elements , we obtain what is called a semi-norm. One can operate with